Computer-assisted techniques for the verification of the Chebyshev property of abelian integrals
In [\textit{M. Grau} et al., Trans. Am. Math. Soc. 363, No. 1, 109--129 (2011; Zbl 1217.34052)], a criterion ensuring that a finite set of abelian integrals has the Chebyshev property was provided. This criterion is based on proving that certain functions, namely the Wronskians of balanced functions of the integrands, have no zero on a real interval. When the involved functions are algebraic, this criterion can be effectively applied by using resultants. The present paper describes a technique to prove that a real analytic function has no zeros on a real interval using interval arithmetic and a rigorous computer analysis, see [\textit{W. Tucker}, Validated numerics. A short introduction to rigorous computations. Princeton, NJ: Princeton University Press (2011; Zbl 1231.65077)]. This technique can be applied to analytic functions which are either algebraic or not algebraic. Therefore, a method to effectively apply the aforementioned criterion in any case is provided in this manuscript. In the paper [\textit{F. Dumortier} and \textit{R. Roussarie}, Discrete Contin. Dyn. Syst., Ser. S 2, No. 4, 723--781 (2009; Zbl 1186.34080)], a conjecture about the Chebyshev property for abelian integrals involving real polynomials and exponential functions was formulated. By combining both the sufficient criterion and the interval arithmetic technique, the authors are able to prove this conjecture for the first three abelian integrals.
scientific article; zbMATH DE number 6154875